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Full text of "Treatise On Analysis Vol-Ii"

11    REGULARIZATION        301

(b)    With the help of Problem 10, construct a sequence (w() of continuous positive
functions, with supports contained in V2, and such that || wt\\ ^ 1 for all /, the sequence
(wt) being uniformly bounded and equicontinuous and such that the set of functions
w,(YO*>'i — wj) is uniformly bounded as zt runs through Ut for each / ^> 1 .

(c)    With the help of Problem 9, construct a sequence (gf) of positive functions with
supports contained in V4, which is uniformly bounded and equicontinuous and such
that the set of functions rti(^(zi)gi — gt) is uniformly bounded and equicontinuous as
zi runs through U, for each / 2> 1 ; and such that for each function / which is the
uniform limit of a subsequence of the sequence (gt), we have f(x)<f(e) whenever

12. Let G be a locally compact group with no small subgroups, and let V be a symmetric
compact neighborhood of>, containing no subgroup of G other than {e}, and such that
for x, y e V the relation x2 = y2 implies that x = y (Section 12.9, Problem 6(a)). A
continuous homomorphism rh~* X(r) of R into G is called a one-parameter subgroup
of G (Section 12.9, Problem 7). The constant one-parameter subgroup r\— *e is de-
noted by 0. For each / e R, the one-parameter subgroup r\— *X(tr) is denoted by tX.
If X is a one-parameter subgroup of G, a continuous real-valued function / on G
is said to be X-differentiable if the function r~l(^((X(r))f—f) converges uniformly
on G to a limit D*/ as r-^-0 in R. If so, then / is also /JSf-differentiable for all
real t, and we have T>txf= tDxf. If /is Jf-differentiable, then for each x e G the
function r\-+f(X(—r)x) is differentiate on R, and its derivative is the function

A continuous bounded real-valued function /on G is said to be adequate if there
exists a neighborhood V of e in G such that || Y(z)f— f\\ =£ 0 for all z e V. The functions
/constructed in Problem ll(c) as limits of subsequences of the sequence (gf) are
adequate. Show that, if /is adequate and if D*/ exists for some one-parameter sub-
group X T£ 0, then Dxf^ 0. If on the contrary/is not adequate, then there exists a one-
parameter subgroup X ^ 0 such that D*/= 0. (Consider a sequence (a,) of elements
^e in G, tending to e, and such that ||Y(a/)/— /ll = 0- Show that it can be assumed
that there exists a sequence of integers ra/ with the property that for all r e R the
sequence a^mj^ converges to X(r), where X^Q (Section 12.9, Problem 7).)

13. The notation and assumptions on G and V remain the same as in Problem 12. Let
(fj) be a sequence of bounded continuous functions on G which converges uniformly
on G to a function /. Let (aj) be a sequence of elements of G tending to e, and (mj)
a sequence of integers >0.

(a) Suppose that /is adequate and that the functions mfy(aj)fj —/}) are uniformly
bounded. Then, for each neighborhood U of e, there exists a number e > 0 such that
the relation \r\ < e implies a1™^ e U. (Suppose that U2 c: V. Let

by taking / sufficiently large, we may suppose that a/ e U and \\y(z)fj —fj\\ ^ j8 > 0
for all z 6 V — U. Show that we cannot have a] e V — U for k < <xmj//3, by using the
inequality

(b) Suppose that the functions mj(j(aj)fj - fj) are uniformly bounded and uniformly
equicontinuous, and also that for each r e R the sequence (a^mj]) tends to X(r)t
where X is a one-parameter subgroup. Show that Dxf exists and is the limit of the - e for